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I'm working in a project that uses"blade solidity ratio", or "rotor solidity", the problem is, there are several articles which defines the equation in a different way changing in the section of the Radius analysis. I found 3 equations that I will describe below:

$$1. \frac{Nc}{2πR}$$ $$2. \frac{Nc}{2R}$$ $$3. \frac{Nc}{R}$$ N=Number of Blades, c= Chord Length, R= Radius of the rotor

Which one is correct or there is some case where each of this equation is applied? Thanks.

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Rotor solidity is the area of the rotor disk that is actually occupied by blade area.

Area of rotor blades = $N \cdot c \cdot R$

Area of rotor disk = $\pi \cdot R^2$

$$Rotor \ solidity = \frac{N \cdot c}{\pi \cdot R} \tag{4}$$

So all three of the equations OP found would not be usual. Good reference books are:

  • Helicopter Performance, Stability and Control by Raymond R. Prouty
  • Principles Of Helicopter Aerodynamics by J. Gordon Leishman.

Both these books use equation (4) for rotor solidity. It is possible for a factor 2 to pop up in the coefficients which are usually considered regarding disk area and solidity: $C_T, C_P$ and $C_Q$. From Leishman:

enter image description here

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  • $\begingroup$ This is generally correct, but if a paper defines it differently and then presents results with respect to it, then that definition should be used for those results/formulae. I've also seen different definitions of solidity (usually for aeroplane propellers though), presumably to account for typical blade shapes etc. $\endgroup$
    – Zeus
    Jul 17, 2019 at 0:34
  • $\begingroup$ Thanks for your response, this was very helpful. $\endgroup$
    – Resou
    Jul 17, 2019 at 15:10
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Dimensionally, those expressions are the same (and thus comparable), except of course for the $2\pi$ constant.

Check the rest of the equations, it is very likely the numeric factor is simply elsewhere.

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I would say that each article with expressions should be analyzed separately, because if the cross-flow turbine is viewed from the upwind side then it covers for wind a rectangular area corresponding to pseudo solidity at expression 2, but considering that at crossflow turbines down the wind side of this area are the same blades obstructed the same rectangular area, then the solidity meets 2 times higher, which corresponds to expression 3.

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